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MATHS 361:  Partial Differential Equation

Tutorial 8:  Fourier transforms

1. Use the definition of Fourier transforms to show that all even functions have a purely real Fourier transform.

2. Consider the‘sgn’function.  sgn(t) = 1 if t > 0 and sgn(t) = − 1 if t < 0.  Can you calculate the Fourier transform of sgn(t) directly? What limit could you take in order to calculate F(sgn(t))?

3. Use Fourier transforms to solve the partial differential equation:

utt = 4uxx ,u(x,0) = e−|x |,ut(x, 0) = 0

You may assume u → 0 at infinity for all t

4. Use Fourier transforms to solve the Partial differential equation

ut+^tux = 0,u(x,0) = H(x + 1)H(1 − x)

5. Use Fourier transforms to solve the Partial differential equation:

ut = uxx ,u(x, 0) = e −x2   = f(x)

.

6. Bonus question:

Take your answer to the previous questions, and think what happens when you run your equation backward in time.  IE, what does u(x,t) looks like for t = −1, t = − 10 and so on.  Do your answers  always make sense? When do you run into trouble?

7. Reflections

I have the equation

uxx + uyy = 1,u(x,0) = δ(1 − x), 0 < x,y

With the assumption that u → 0 and x or y approach infinity.

(a) Suppose I told you that u(0,y) = 0. How would extend the above PDE in order to give a PDE

over −∞ < x < ∞ .

(b) Suppose I told you that ux (0,y) = 0. How would extend the above PDE in order to give a PDE

over −∞ < x < ∞ .

(c) Hard.   Bonus Question.   Consider attacking the rest of the tutorial first.  Solve the u(0,y) = 0 case using fourier transforms. It may be useful to note that F ( sgn(t)) =